Will a bad dryer stop the compresses from coming on?

There isn't a lot that can go wrong with a receiver/dryer in an automobile air conditioning system, unless you have damaged it. That shouldn't stop the compressor clutch from engaging. The sensor that is usually on the side of the receiver/dryer detects the pressure of the refrigerant. If the pressure drops below a certain point, the sensor will not allow the clutch to engage. It's possible that the receiver/dryer has a hole in it, but that's usually caused by damage, not by normal use. However, the clutch itself can go bad, since it works with a magnetic coil that can burn out. You can check that by following the wire coming out the back of the clutch (behind the pulley) up to the nearest connector and testing it there with an ohmmeter. If it is open, you'll need to pull the pulley to replace. Better take it to a air conditioning shop.